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- chance constraints (2)
- Bilevel optimization (1)
- Black-box constraints (1)
- Chance constraints (1)
- Cutting planes (1)
- European gas market (1)
- Mathematical Programs with Equilibrium Constraints, probabilistic con- straints, M-stationarity conditions, calmness (1)
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We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.
Value at risk approach to producer's best response in electricity market with uncertain demand
(2021)
We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided.
Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.
We consider Mathematical Programs with Equilibrium Constraints with proba-
bilistic constraints (PMPECs). Such models have proven to be useful in modeling
electricity or gas markets subject to random parameters. Our main interest is the
derivation of Mordukhovich (M-) stationarity conditions under suitable constraint
quali...cations ensuring the calmness of the canonically perturbed generalized equation.
Applying recent results from deterministic MPECs, we identify the needed properties
of the probability function in order to derive explicit M-stationarity conditions. The
results are applied to a simple stochastic bilevel problem in an economic context.
In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are
optimal control problems related to storage management systems with uncertain in- and output. We give suffcient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. Hence we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics.